Tight bounds towards Zarankiewicz problem in hypergraph

Abstract

The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\dots ,s_r}$ be the complete $r$-partite $r$-graph such that the $i$-th part has $s_i$ vertices. We say an $r$-partite $r$-graph $H=H(V_1,\dots ,V_r)$ contains an ordered $K_{s_1,\dots ,s_r}$ if $K_{s_1,\dots ,s_r}$ is a subgraph of $H$ and the set of size $s_i$ vertices is embedded in $V_i$. The Zarankiewicz number for $r$-graph, denoted by $z(m_1,\dots ,m_r;s_1,,\dots ,s_r)$, is the maximum number of edges of the $r$-partite $r$-graph whose $i$-th part has $m_i$ vertices and does not contain an ordered $K_{s_1,\dots ,s_r}$. In this paper, we show that $$z(m_1,m_2,\cdots ,m_{r-1},n;s_1,s_2,\cdots ,s_{r-1},t)=\hat{I}˜\left(m_1{m}_2\cdots m_{r-1}{n}^{1-1/s_1{s}_2\cdots s_{r-1}}\right)$$ for a range of parameters. This extends a result of Conlon [Math. Proc. Camb. Philos. Soc. (2022)].

10 pages